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Edited by: Laurent Petit, Centre National de la Recherche Scientifique (CNRS), France

Reviewed by: Karla Miller, University of Oxford, United Kingdom; Caroline Magnain, Massachusetts General Hospital, Harvard Medical School, United States

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

3D-Polarized Light Imaging (3D-PLI) enables high-resolution three-dimensional mapping of the nerve fiber architecture in unstained histological brain sections based on the intrinsic birefringence of myelinated nerve fibers. The interpretation of the measured birefringent signals comes with conjointly measured information about the local fiber birefringence strength and the fiber orientation. In this study, we present a novel approach to disentangle both parameters from each other based on a weighted least squares routine (ROFL) applied to oblique polarimetric 3D-PLI measurements. This approach was compared to a previously described analytical method on simulated and experimental data obtained from a post mortem human brain. Analysis of the simulations revealed in case of ROFL a distinctly increased level of confidence to determine steep and flat fiber orientations with respect to the brain sectioning plane. Based on analysis of histological sections of a human brain dataset, it was demonstrated that ROFL provides a coherent characterization of cortical, subcortical, and white matter regions in terms of fiber orientation and birefringence strength, within and across sections. Oblique measurements combined with ROFL analysis opens up new ways to determine physical brain tissue properties by means of 3D-PLI microscopy.

Understanding the human brain's function and dysfunction requires a thorough knowledge about the brain's fiber tracts, forming a dense network of connections within, but also between the different brain regions. Over the last years, several imaging techniques have emerged which are capable of resolving anatomical structures with different spatial resolutions. At the millimeter scale, diffusion MRI is the most prominent one as it is applicable to both

3D-Polarized Light Imaging (3D-PLI) (Axer et al.,

A pitfall of any histological imaging technique is the extraction of information in the direction perpendicular to the sectioning plane, i.e., in the depths of the histological section. While in-plane fiber orientations can be obtained from texture information of microscopic images via the structure tensor without the need for any biophysical model (Budde and Frank,

As an alternative, 3D-PLI enables to determine fiber orientations directly from the measurement data without the need for a manual choice of image processing parameters. This is achieved by taking measurement data from different views by tilting the brain section and analyzing it with a biophysical model (Axer et al.,

Several algorithms for the analysis of the data acquired with the tiltable specimen stage have been presented so far. The Markov Random Field algorithm by Kleiner et al. (

The here presented approach seeks to overcome this noise instability and provide a more robust separation of fiber density and orientation. Our approach utilizes a weighted least squares algorithm to process the tilted measurements. It is then compared to the analytical DFT algorithm on synthetic and experimental data. The examined experimental results represent the first analysis of large-scale 3D-reconstructed human brain data in 3D-PLI. The new approach was introduced in Schmitz et al. (

3D-PLI utilizes the birefringence of nerve fibers which is measured in customized polarimeters. The birefringence originates from the regular arrangement of lipids in the myelin sheath resulting in optical anisotropy. This anisotropy causes a phase shift of incident polarized light passing the brain tissue. The optical setup as described in Axer et al. (

Experimental setup and tilting coordinate system.

The effective physical model behind 3D-PLI describes myelinated nerve fibers as negative uniaxial birefringent crystals. Assuming ideal optical components, this model yields the following expression of the light intensity in each pixel during rotation of the filters (Axer et al.,

_{rel} in contrast to Axer et al. (_{s}, birefringence Δ_{T} is given by the average of the profile, the direction angle by its phase and the retardation sinδ by its relative amplitude (Axer et al.,

A mechanical solution to gain more measurement information is a tiltable specimen stage built into the polarimetric system, which introduces a pre-defined tilting angle to the nerve fiber orientation relative to the sectioning plane (Axer et al.,

For an accurate model of the data acquisition, the noise level of the imaging system must be taken into account. As the detected signals are light intensities captured by a CCD camera in our case, the dominant noise sources are shot noise during image acquisition and the internal signal processing of the camera (Bertolotti, ^{2} on the number of detected photons as the expected value μ can only be determined experimentally. Therefore, 100 samples of the same scene were taken and analyzed pixelwise for the variances and expected values of the measured light intensities. The resulting relationship between variance and expected value is given by σ^{2} = 3μ (cf. ^{2} =

We will use the coordinate system introduced by Kleiner et al. (_{Tilt} as the number of tilting positions the tilting directions equidistantly spanning the space are in general given by _{t}_{t}

We introduce the indices _{T} = _{Tilt} + 1 and the number of polarizer positions acquired as _{P}. All measurement data accumulates to _{T} · _{P} light intensities _{ji} for each pixel. Application of the Fourier analysis to all measurements as described in Axer et al. (_{T} transmittance values _{j, T}, _{T} retardation values |sin δ_{j}| and _{T} direction values φ_{j}. In this notation the intensity curve of a measurement can than be expressed as
_{ji} predicted by the 3D-PLI model are given by inserting Equation (7) into Equation (8):
_{N} can now be formulated as a weighted least squares problem. With the weights

A starting point for the direction angle is given by the direction angle derived from the planar measurement φ_{0}. For the inclination α and the relative thickness _{0}, α_{l}, _{k}],

The boundaries of the parameter space could be accounted for by utilizing an optimization algorithm capable of dealing with hard boundaries. Yet, in our case a more elegant solution is to exploit the symmetry of the problem. This enables to reformulate the bounded optimization problem as an unbounded problem. As _{u}, α_{u}) can be transformed back into the standard 3D-PLI parameter space by the following transformations:

Pseudocode of the ROFL algorithm

For this study, the ROFL algorithm was implemented in ^{6} pixels). Therefore, the algorithm was parallelized pixelwise using

Monte-Carlo simulations were carried out to test the robustness of the presented approach and the DFT algorithm against measurement noise.

Simulations can provide information about two problems. Firstly, the accuracy of the obtained parameters can be compared to the synthetic ground truth. Secondly, the algorithms can be tested against biases in their parameter estimation. One sythetic dataset each was created to answer these questions. For both datasets, the generation of synthetic 3D-PLI signals for a specific parameter set is executed in the same way.

Synthetic signals were generated in a similar manner as in Wiese et al. (_{g} with the parameters (_{g}, φ_{g}, α_{g}) was rotated into four tilting positions (North, South, East, West) by τ = 5.51°, the assumed internal tilting angle for human brain tissue. For each tilting position a sinusoidal light intensity profile according to Equation (1) was calculated. For the transmittance _{T} 2500 was chosen, as it represents a typical transmittance value for human brain sections. To mimic the experimentally observed light intensity distribution, for each calculated light intensity _{noisy} was then computed by drawing one sample from a negative binomial distribution with expected value given by

In a first simulation, the accuracy of the obtained parameters was assessed. As our primary interest here was the validation of the reconstruction of the inclination and the relative section thickness, only one direction angle, φ = 45°, was simulated. This direction angle also represents the worst case scenario for the four tilting positions as the angle between the orientation vector and the rotated orientation vector is maximal for φ = ψ and decreases with |φ − ψ|. With respect to the inclination angle, it is sufficient to simulate only positive inclinations as the inclination sign does not effect the reconstruction precision. As ground truth vectors all combinations of the fixed direction angle 45°, inclinations α from 0°, 1°, …, 89°, 90°, and thicknesses

To tests the algorithms against biases in their inclination estimation, a second dataset of 500,000 samples of orientation vectors uniformly distributed on the unit sphere were computed. From these vectors, a dataset of direction and inclination angles was derived and used to calculate the noisy sinusoidal light intensity profiles with a relative thickness of

The simulated sinusoidal profiles were analyzed with the ROFL algorithm and the DFT algorithm resulting in the reconstructed vector _{r} with the parameters (_{r}, φ_{r}, α_{r}). The accuracy of the obtained orientation was then measured by the acute angle γ between the ground truth vector _{g} and the reconstructed vector _{r}:
_{d}〉.

To assess if the obtained inclinations are biased, inclination angles were reconstructed from the second synthetic dataset. The resulting inclination angle histograms were then compared to the ground truth inclination histogram.

To test and compare the DFT and the ROFL algorithm on experimental data, a series of coronal sections of a right human hemisphere was investigated. The post mortem human tissue sample used for this study was acquired in accordance with the local ethic committee of our partner university at the Heinrich Heine University Düsseldorf. As confirmed by the ethic committee, postmortem human brain studies do not need any additional approval, if a written informed consent of the subject is available. For the research carried out here, this consent is available.

The examined brain was removed within 24 h after the donor's death. The right hemisphere was fixed in 4% buffered formaldehyde solution for 15 days to prevent tissue degeneration. After immersion in a 20% solution of glycerin with Dimethyl sulfoxide (DMSO) for cryoprotection the brain was frozen at a temperature of −80°C. The coronal cutting plane was decided to be orthogonal to a line connecting the anterior and posterior commissures. Before sectioning, the frontal lobe was cut off to create a stable cutting platform. The sectioning resulted in 843 sections of 70 μm thickness (

After an intensity based calibration as described in Dammers et al. (^{2}. As a comparison, the DFT algorithm was applied to determine the inclination angle and relative section thickness. Also, the residuum was calculated for the parameters obtained by DFT.

After the sectionwise analysis, the 3D-reconstruction of the measured sections requires a multi-step registration to retain the 3D volume of the measured brain. Here the blockface volume generated by aligning the blockface images to a 3D volume serves as a reference for the registration of the histological sections (Schober et al.,

Traditional anatomical studies have described the fiber pathways globally and without considering differences between brains. At the level of detail presented in this study with a voxel size of 64 × 64 × 70μm^{3}, the inter-subject variability in the fine structure of fiber orientations becomes much more relevant. Therefore, the resulting fiber orientations cannot easily be compared to anatomical atlases. A complementary measurement at the same resolution enabling a comparison is not available either. Additionally, a phantom for 3D-PLI has not been developed yet, impeding the possibility of a measurement with a known ground truth. Therefore, we chose to validate the resulting orientations based on their coherence across the whole volume and their alignment to anatomical boundaries, for example, within the sagittal stratum. The results of both algorithms were furthermore compared from a statistical point of view by the difference between the predicted and measured light intensities measured by the sum of the squared residuals.

The simulation results of the first simulated dataset are depicted in Figure

Simulation results. _{d}〉 as a function of relative thickness

The orientation reconstruction accuracy of the DFT is valley shaped with respect to the inclination angle and becomes minimal for an inclination angle of α ≈ 60°. From this minimum the orientation reconstruction error increases slightly for flat fibers. The most challenging orientations to analyze are very steep fibers with respect to the sectioning plane: for these, the orientations predicted by the DFT algorithm differ strongly from the ground truth. Even for relative thicknesses of

In contrast, the orientation reconstruction error for the ROFL algorithm does not take the form of a valley: for all section thicknesses

Similar to the the orientation reconstruction error, the relative reconstruction error 〈σ_{d}〉 of the relative thickness _{d}〉 > 1 for both algorithms. For very steep fibers, the error observed in the results of the DFT algorithm increases to up to 80%. The ROFL algorithm achieves a lower error in this case, yet the best case for

As the orientation vectors were distributed uniformly on a sphere, the frequency of the ground truth inclinations is proportional to the circumference of the cross section of the sphere at the respective height corresponding to a given inclination. In our definition of the inclination angle as

The inclination histograms obtained from the simulated dataset of uniformly distributed orientation vectors are depicted in Figure

Inclination histograms for uniformly distributed orientations obtained from ROFL and DFT algorithms and ground truth. The magenta line depicts a fit curve to the ground truth inclination histogram. Fit result:

To demonstrate the working principle of the ROFL algorithm by means of experimental data, one pixel of the stratum sagittale highlighted in Figure

Working principle of the ROFL algorithm demonstrated for a single pixel. ^{2} = 0.97.

The boundaries of the investigated 230 coronal sections are illustrated in the view of the blockface volume shown in Figure

Reconstructed 3D-PLI volumes.

Volumetric views of 3D-PLI modalities are shown in Figure

Reconstructed 3D-PLI modalities.

Major differences between the ROFL algorithm and the DFT algorithm were particularly observed for two cases: low relative section thickness and very steep fibers with respect to the sectioning plane. Therefore, the vector fields of two regions were investigated more closely: the stratum sagittale which is expected to run perpendicular to the sectioning plane and a region at the boundary of white and gray matter. The vector fields underlaid with the retardation maps are shown in Figure

Vectorfields obtained with the ROFL algorithm (left) and the DFT algorithm (right) underlayed with the retardation map.

3D-PLI modalities of a representative brain section.

In addition to the potentially biased visual inspection of the vector fields, an unbiased statistical statement about the reconstruction accuracy is given by the mean residual: the mean value of the mean residuals of all pixels is 15% lower for the results of the ROFL algorithm than the results of the DFT algorithm.

Since the ROFL algorithm performs a least squares fit in all image pixel, the actual sum of residuals χ^{2} (as a result of the optimization process) was accessible. For the DFT algorithm the residual map can also be calculated, yet this requires additional computations which take even longer than the runtime of the algorithm itself. Figure ^{2} (Figure ^{2} is a measure of the difference between the model and the experimental data, it is expected to increase for artifacts such as dust particles which rotate with the polarization filters. Such artifacts were indeed observed in experimental results (cf. Figure ^{2} measure, however, is sensitive to such artifacts.

Identification of artifacts and fiber crossings. ^{2} with green arrows indicating dust particles.

In addition, we found areas of increased residuals (cf. red arrows in Figure

We introduced the least-squares algorithm ROFL for the reconstruction of fiber orientation and the extraction of the relative section thickness from measurements with a tiltable specimen stage in 3D-Polarized Light Imaging (3D-PLI). This method requires only one additional assumption besides the polarimetric model: the refractive index of the brain tissue. This represents a substantial improvement as compared to other histological imaging techniques which strongly rely on parameter dependent image processing pipelines to extract three-dimensional fiber orientations. To our knowledge, no other histological imaging technique is currently capable of deriving three-dimensional information from a biophysical model. This compensates for the disadvantage of the difficulties accompanying the 3D re-alignment of serial high-resolution brain section images as required for 3D-PLI anaylsis.

The working principle of the ROFL algorithm was proven for simulated data for which it resulted in a significant improvement of the orientation reconstruction. It was opposed to a previously implemented analytical algorithm based on a discrete Fourier transformation (DFT). The reconstruction with ROFL became more reliable, in particular for low relative section thicknesses. The noise instability of the DFT algorithm for very flat fibers was already observed by Wiese et al. (

ROFL was applied to 230 consecutive human brain sections subjected to 3D-PLI. The results were 3D reconstructed to demonstrate the robustness and reproducibility of our approach. The obtained modality volumes of retardation, relative section thickness and fiber orientation were characterized by high coherence across the sections. Anatomical structures, such as fiber tracts, white/gray matter borders, vessels could be reconstructed with high precision. The transmittance volume displayed brightness variations, yet the strength of our approach is that the determination of the fiber orientation and relative section thickness is independent of the transmittance due to the normalization. By eyesight, no major differences were observed in the vector fields resulting from the ROFL and DFT approaches. In the stratum sagittale, both approaches resulted in very similar orientations with very minor differences. The computed inclinations were on average α ≈ 65° and the computed relative section thicknesses

The inclination histograms, however, yielded a very important finding: whereas, the frequency of the computed inclination angles decreased significantly for in-plane fibers with α ≈ 0° for the DFT algorithm, the ROFL algorithm was still capable of reconstructing the whole spectrum of possible inclinations. The same behavior was observed for simulated data, which proves a systematic bias of the DFT algorithm. The ROFL result is also more plausible as for the observed ROI in Figure

Furthermore, the ROFL algorithm enables a direct way to evaluate the agreement of model and data based on the residuum map which would otherwise have to be computed additionally. This kind of map was exploited to identify measurement artifacts and, even more importantly, crossing fiber regions which were not classifiable by previous 3D-PLI analysis approaches (Axer et al., ^{3}. A voxel containing fibers with different courses (i.e., orientations) will inevitably result in a 3D-PLI measurement composed of superimposed birefringence signals. This might lead to a biased orientation interpretation. To give an example, a voxel comprising crossing out-of-plane and in-plane fibers, the ROFL algorithm will preferably reconstruct the orientation which causes a larger signal, in this case the in-plane orientation. To overcome this issue, different strategies are currently being explored: (i) Modeling of crossing fiber structures and subsequent simulations of tilting measurements utilizing the simPLI simulation platform introduced by Dohmen et al. (

While the obtained parameters show a high coherence across the volume, no quantitative statement about the reliability of the resulting parameters is possible at this point. Hence, future studies need to investigate the uncertainty of the fitted parameters. For Diffusion Tensor Imaging, for example, bootstrapping approaches were used to obtain orientation confidence maps (Jones,

The improved reconstruction comes at the cost of computation time, which increases by a factor of 5,000. Yet, as processing a single brain section takes about 3 min using four compute nodes on JURECA (Jülich Supercomputing Centre,

To conclude, the present approach has opened up a new way to determine physical tissue properties from oblique measurements in microscopic 3D-PLI. Cortical, subcortical, and white matter regions could be characterized coherently across brain sections in terms of fiber orientation and birefringence strength. This is a prerequisite for subsequent volume-based connectivity analysis.

DS: study design, algorithm implementation, simulation, processing of experimental data, evaluation, writing, discussion. SM and MS: 3D reconstruction, discussion. NS: 3D visualization. MM: provided the brain sample and anatomical interpretations. KA and TL: writing, discussion. MA: study design, discussion, writing.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We would like to thank Marcus Cremer and Patrick Nysten for the preparation of the examined brain sections and Felix Matuschke for fruitful discussions about the statistical analysis. We also thank David Gräßel who significantly helped in identifying anatomical structures. The authors gratefully acknowledge the computing time granted by the JARA-HPC Vergabegremium and provided on the JARA-HPC Partition part of the supercomputer JURECA at Forschungszentrum Jülich.

The Supplementary Material for this article can be found online at: